Type Theory in Rosario Thorsten Altenkirch July 25, 2011 Thorsten - - PowerPoint PPT Presentation

Type Theory in Rosario

Thorsten Altenkirch July 25, 2011

Thorsten Altenkirch () Rosario July 25, 2011 1 / 7

Zermelo-Fraenkel Set Theory

Zermelo (1871-1953) Fraenkel (1891-1965) Axiomatic Set Theory ≈ 1925 ZFC = Zermelo-Fraenkel with Axiom of Choice Foundations of modern Mathematics Additional axioms, e.g. the continuum hypothesis

Thorsten Altenkirch () Rosario July 25, 2011 2 / 7

Set Theory for Computer Science?

Set Theory is untyped (everything is a set), while programming languages are typed (either statically or dynamically). Basic concepts from computer science (records, functions) are not primitive in Set Theory. Basic operations in set theory (e.g. ∩ , ∪) are not directly available

• n types.

Set Theory is not constructive, i.e. there is a set theoretic function solving the Halting Problem.

Question:

Is there an alternative to Set Theory?

Thorsten Altenkirch () Rosario July 25, 2011 3 / 7

Martin-Löf Type Theory

Per Martin-Löf (1942-) Martin-Löf introduced Type Theory as a constructive foundation of Mathematics since 1972. Type Theory doesn’t rely on predicate logic but uses types to represent propositions. Basic operations on types are Π-types (dependent function types) and Σ-types (dependent records). Type Theory is a programming language.

Thorsten Altenkirch () Rosario July 25, 2011 4 / 7

Propositions as types (The Curry-Howard Isomorphism)

A proposition corresponds to the types of it proofs. A proposition is true if the corresponding type is non-empty. Conjunction A ∧ B is repesented by cartesian product (A × B). Implication A → B is represented by function types A → B (looks the same). ∀ and ∃ correspond to Π (depednent function) and Σ (dependent records).

Thorsten Altenkirch () Rosario July 25, 2011 5 / 7

Agda

Ulf Norell Ulf Norell has implemented Agda, a functional programming language based on Type Theory in his PhD in 2007. Agda is inspired by earlier systems such as Epigram, Cayenne and Coq. Agda can be used to program and to reason.

Thorsten Altenkirch () Rosario July 25, 2011 6 / 7

Course contents

Monday Agda intro, Programming with dependent types Tuesday Propositions as types Wednesday Mixing programming and reasoning Thursday Inductive relations, proof trees Friday Safe evaluation and typed assembly language Material, links, exercises: http://www.cs.nott.ac.uk/~txa/rosario/

Thorsten Altenkirch () Rosario July 25, 2011 7 / 7